The present invention relates to a sample analysis method which enables analyzing a sample provided with a dielectric film formed of a ferroelectric substance or a high-dielectric constant substance, by an ellipsometer in overall optical range by forming a model including a great amount of void based on effective medium approximation.
Conventionally, an ellipsometer has been used for analyzing information on a sample. An ellipsometer is designed to obtain a phase difference (Δ: delta) and an amplitude ratio (Ψ: psi) by applying polarized light to a sample and measuring a difference between a polarization state of the incident light and that of the reflected light. The sample to be analyzed includes a simple substrate, a substrate having a film formed thereon, and the like.
FIG. 9 is a graph showing the measurement result, of a sample provided with a silicon dioxide film formed on a silicon substrate, by an ellipsometer. The abscissa axis denotes the wavelength (nm: nanometer) of the incident light into the sample, the right ordinate axis denotes the measured phase difference (Δ: delta), and the left ordinate axis denotes the measured amplitude ratio (Ψ: psi). It should be noted that, when the wavelength of the incident light is 633 nm, the dielectric constant of the silicon dioxide film, based on optical measurement, is approximately 2.1.
A single combination of a refractive index (n), an extinction coefficient (k) and a thickness (d) of a film cannot be obtained directly from a phase difference and an amplitude ratio of a sample measured by an ellipsometer as described above. Accordingly, in order to obtain the single combination of the refractive index, the extinction coefficient and the thickness of a film from the measured phase difference and amplitude ratio, formation of a model, based on some combinations (n, k, d), corresponding to the sample and comparison between a phase difference and an amplitude ratio obtained theoretically from the model and a phase difference and an amplitude ratio measured by an ellipsometer are performed in addition to the measurement using the ellipsometer. It should be noted that the formation of a model includes setting of conditions corresponding to the physical properties of the sample, and the items of the conditions to be set include the material of the substrate and the film, the thickness of each film layer, and the optical constants of the substrate and the film. Moreover, used usually for setting each item are a known reference corresponding to the sample, a required dispersion formula which represents the wavelength dependence of the dielectric constant and has a plurality of parameters, or other particular matters.
Furthermore, a process (which will be referred to as fitting) is performed to change the parameters of the dispersion formula, the thickness of each film layer of the model, or other matters so that the degree of the difference between the both to be obtained from the above comparison becomes minimal. As a result of fitting of the difference between the both, usually obtained by calculation using the least squares method, is determined that the result obtained by the least squares method, has become relatively small, the refractive index and the extinction coefficient of the film are obtained from the values of the parameters of the dispersion formula, and the thickness is specified as the thickness of the film in the sample. It should be noted that the formation of a model, the calculation which uses the least squares method, the fitting and the like are generally performed manually or automatically based on a desired program using a computer (see Japanese Patent Application Laid-Open No. 2002-340789 and Japanese Patent Application Laid-Open No. 2002-340528).
The analysis method described above can be performed without trouble when the boundary between a substrate S1 and a film S2, which constitute a sample S as shown in FIG. 10A, is flat, the film S2 is smooth and the substance constituting the film S2 is homogeneous and continuous. However, since an actual sample S has roughness on the surface of the film S2 as shown in FIG. 10B, a favorable result may not be obtained by performing the above analysis method without modification, depending on the degree of the roughness.
Accordingly, in such a case, a model is formed using an idea of effective medium approximation which substitutes the film S2 having roughness shown in FIG. 10B with a first film S2a, which is formed of a homogeneous and continuous substance of the film S2 and has a thickness d1, and a second film S2b, which is formed of a substance M of the film S2 including a desired ratio of void V (percentage of void, which will be referred to as a void) and has a thickness d2. Sample analysis is made by applying fitting to the thickness d1 of the first film S2a, the thickness d2 of the second film S2b, the volume fractions of the material M and the void in the second film S2b, and the parameters of the dispersion formula corresponding to the sample with respect to the model formed as described above.
The value of the void to be set for a model is decided based on the degree of the roughness, and the maximum value of the void is generally considered to be approximately 40˜55% in view of the range which enables formation of a film and often is set to be 50%. It should be noted that, when it is unknown whether the film surface of the sample has roughness or not, it is generally determined whether roughness exists or not by performing both analyzing methods, one which uses effective medium approximation mentioned above, and another which does not use effective medium approximation, and judging which analyzing method gives a result more highly compatible to an actual sample, using the calculation result of least squares method.
Moreover, the effective medium approximation is applied not only to a layer of a case where the film surface of a sample has roughness, but also to a boundary layer of a case where the boundary between a substrate and a film or the boundary between film layers has roughness. Furthermore, the effective medium approximation may be used for decreasing the value of the refractive index as a technique for actually making the analysis, regardless of existence of roughness. In this case, whether effective medium approximation is to be used or not is also judged by determining the result of analysis using a model including void based on the effective medium approximation.
For example, assume an analysis of a sample in which a first film made of amorphous silicon is formed on a glass substrate and a second film made of a native oxide film is formed on the first film. Further assume that the calculation result of the least squares method (which corresponds to the mean square error χ2) becomes 16.6 when the thickness of the first film of this sample is first set to be 2,000 angstroms, the thickness of the second film is set to be 20 angstroms, a model is formed using a known amorphous silicon reference and fitting is applied to this model.
Next, assume that the calculation result of the least squares method becomes 10.6 and the volume fraction of amorphous silicon in the first film becomes approximately 86% when the thickness of the first film and the thickness of the second film are set to be the same as those described above, the volume fraction of amorphous silicon in the first film is set to be 50% and the void of the first film is set to be 50% using the effective medium approximation regardless of roughness, a model is formed using a known amorphous silicon reference and fitting is applied in the same manner as described above. Since a smaller value of the calculation result of the least squares method is preferable, it is found that the result of analysis (result of fitting) obtained by using a known reference and by forming a model including void based on effective medium approximation is more preferable than that of the first case.
Finally, assume that the thickness of the first film and the thickness of the second film are set to be the same as those of the first case, the volume fraction of amorphous silicon in the first film is set to be 86% and the void of the first film is set to be 14% using the effective medium approximation regardless of roughness in view of the volume fraction of amorphous silicon of the second case, a model is formed using not the reference but a dispersion formula and fitting is applied in the same manner as described above. When the calculation result of the least squares method is 0.14 and the volume fraction of amorphous silicon in the first film is approximately 99.16%, it is found that there is no point in formation of a model including void since the void is approximately 0%, although the calculation result of the least squares method is extremely preferable. Accordingly, in such a case, analysis is generally made by forming a model including no void and using a dispersion formula.
A conventional analysis method based on optical measurement using an ellipsometer has a problem that the physical properties of a sample cannot be analyzed based on the complex dielectric constant since the complex dielectric constant may not be obtained in a desired optical range when the sample is provided with a high-dielectric constant film or a ferroelectric film having a dielectric constant equal to or larger than 50 based on electrical measurement.
That is, a range of a conventional analysis method which uses an ellipsometer, is often 248 nm˜826 nm inside overall optical range of light wavelengths (DUV (Deep Ultraviolet)˜NIR (Near Infrared): 190 nm˜1700 nm, which corresponds to 0.75 eV˜6.5 eV of a photon energy range. Each of a high-dielectric constant substance and a ferroelectric substance has three parts of electric polarizability: electric polarizability, ionic polarizability and dipole polarizability. From the optical measurements, it is known that refractive index and extinction coefficient of the ferroelectric and high-dielectric constant substances have a steep peak below 400 nm. However, it is generally considered that the dispersion formula and the reference which can deal with such a sharp data have not been found yet. Accordingly, the fact is that we cannot find any biblio, document or the like which correctly defines how the refractive index and the extinction coefficient of a high-dielectric constant substance and a ferroelectric substance change in the optical range under 400 nm (range of 248 nm˜400 nm). It should be noted that electrical measurement used for measuring a dielectric constant is within a frequency range of 100 kHz˜1 MHz far out of a near-infrared range included in the optical measurement range, and the measurement range of electrical measurement and that of optical measurement are completely different from each other.
Moreover, since the dielectric constant of a sample varies according to the measurement range, the problem described above arises in an optical measurement range under 400 nm. Accordingly, physical properties analysis of a high-dielectric substance or a ferroelectric substance in an optical measurement range under 400 nm should be made using another apparatus which is of great price and requires analysis time.